On Riesz derivative

2019 ◽  
Vol 22 (2) ◽  
pp. 287-301 ◽  
Author(s):  
Min Cai ◽  
Changpin Li
Keyword(s):  
Caputo Derivative ◽  
Fractional Derivatives ◽  
Fractional Laplacian ◽  
One Dimension ◽  
Riesz Derivative ◽  
Liouville Derivative

Abstract This paper focuses on studying Riesz derivative. An interesting investigation on properties of Riesz derivative in one dimension indicates that it is distinct from other fractional derivatives such as Riemann-Liouville derivative and Caputo derivative. In the existing literatures, Riesz derivative is commonly considered as a proxy for fractional Laplacian on ℝ. We show the equivalence between Riesz derivative and fractional Laplacian on ℝn with n ≥ 1 in details.

Mathematical views of the fractional Chua's electrical circuit described by the Caputo-Liouville derivative

2021 ◽  
Vol 67 (1 Jan-Feb) ◽  
pp. 91
Author(s):  
N. Sene
Keyword(s):  
Caputo Derivative ◽  
Local Stability ◽  
Numerical Scheme ◽  
Equilibrium Points ◽  
Electrical Circuit ◽  
Maximal Lyapunov Exponent ◽  
Electrical Circuits ◽  
Adaptive Controls ◽  
Liouville Derivative

This paper revisits Chua's electrical circuit in the context of the Caputo derivative. We introduce the Caputo derivative into the modeling of the electrical circuit. The solutions of the new model are proposed using numerical discretizations. The discretizations use the numerical scheme of the Riemann-Liouville integral. We have determined the equilibrium points and study their local stability. The existence of the chaotic behaviors with the used fractional-order has been characterized by the determination of the maximal Lyapunov exponent value. The variations of the parameters of the model into the Chua's electrical circuit have been quantified using the bifurcation concept. We also propose adaptive controls under which the master and the slave fractional Chua's electrical circuits go in the same way. The graphical representations have supported all the main results of the paper.

scholarly journals Numerical methods for solving the multi-term time-fractional wave-diffusion equation

2013 ◽  
Vol 16 (1) ◽  
Author(s):  
Fawang Liu ◽  
Mark Meerschaert ◽  
Robert McGough ◽  
Pinghui Zhuang ◽  
Qingxia Liu
Keyword(s):  
Numerical Methods ◽  
Theoretical Analysis ◽  
Diffusion Equation ◽  
Fractional Derivatives ◽  
Fractional Laplacian ◽  
Numerical Results ◽  
Diffusion Equations ◽  
Time Space ◽  
Methods And Techniques ◽  
Wave Diffusion

AbstractIn this paper, the multi-term time-fractional wave-diffusion equations are considered. The multi-term time fractional derivatives are defined in the Caputo sense, whose orders belong to the intervals [0,1], [1,2), [0,2), [0,3), [2,3) and [2,4), respectively. Some computationally effective numerical methods are proposed for simulating the multi-term time-fractional wave-diffusion equations. The numerical results demonstrate the effectiveness of theoretical analysis. These methods and techniques can also be extended to other kinds of the multi-term fractional time-space models with fractional Laplacian.

scholarly journals On the Nonlocal Problems in Time for Time-Fractional Subdiffusion Equations

2022 ◽  
Vol 6 (1) ◽  
pp. 41
Author(s):  
Ravshan Ashurov ◽  
Yusuf Fayziev
Keyword(s):  
Fractional Derivatives ◽  
Separable Hilbert Space ◽  
Nonlocal Boundary Condition ◽  
Nonlocal Boundary ◽  
Nonlocal Boundary Value Problem ◽  
Nonlocal Problems ◽  
Existence Of A Solution ◽  
Left Hand ◽  
Liouville Derivative ◽  
The Right

The nonlocal boundary value problem, dtρu(t)+Au(t)=f(t) (0<ρ<1, 0<t≤T), u(ξ)=αu(0)+φ (α is a constant and 0<ξ≤T), in an arbitrary separable Hilbert space H with the strongly positive selfadjoint operator A, is considered. The operator dt on the left hand side of the equation expresses either the Caputo derivative or the Riemann–Liouville derivative; naturally, in the case of the Riemann–Liouville derivatives, the nonlocal boundary condition should be slightly changed. Existence and uniqueness theorems for solutions of the problems under consideration are proved. The influence of the constant α on the existence of a solution to problems is investigated. Inequalities of coercivity type are obtained and it is shown that these inequalities differ depending on the considered type of fractional derivatives. The inverse problems of determining the right-hand side of the equation and the function φ in the boundary conditions are investigated.

Dispersion analysis for wave equations with fractional Laplacian loss operators

2019 ◽  
Vol 22 (6) ◽  
pp. 1596-1606
Author(s):  
Sverre Holm
Keyword(s):  
Phase Velocity ◽  
Fractional Derivative ◽  
Fractional Derivatives ◽  
Wave Equations ◽  
Fractional Laplacian ◽  
Velocity Dispersion ◽  
Dispersion Analysis ◽  
Phase Velocity Dispersion ◽  
Loss Term ◽  
Kronig Relation

Abstract Several wave equations for power-law attenuation have a spatial fractional derivative in the loss term. Both one-sided and two-sided spatial fractional derivatives can give causal solutions and a phase velocity dispersion which satisfies the Kramers–Kronig relation. The Chen–Holm and the Treeby–Cox equations both have the two-sided fractional Laplacian derivative, but only the latter satisfies this relation. There also exists several seemingly different expressions for the phase velocity for these equations and it is shown here that they are approximately equivalent. Causality of the Chen–Holm equation has also been a topic of some discussion and it is found that despite the lack of agreement with the Kramers–Kronig relation, it is still causal.

scholarly journals A new second-order midpoint approximation formula for Riemann–Liouville derivative: algorithm and its application

2017 ◽  
Vol 82 (5) ◽  
pp. 909-944 ◽  
Author(s):  
Hengfei Ding ◽  
Changpin Li ◽  
Qian Yi
Keyword(s):  
Fractional Derivatives ◽  
Second Order ◽  
Stability And Convergence ◽  
Approximation Formula ◽  
Finite Difference Schemes ◽  
First Order ◽  
Liouville Derivative ◽  
Two Space Dimensions ◽  
Fractional Differential ◽  
Nicolson Scheme

Abstract Compared to the classical first-order Grünwald–Letnikov formula at time $t_{k+1}\; (\text{or}\; t_{k})$, we firstly propose a second-order numerical approximate formula for discretizing the Riemann–Liouvile derivative at time $t_{k+\frac{1}{2}}$, which is very suitable for constructing the Crank–Nicolson scheme for the fractional differential equations with time fractional derivatives. The established formula has the following form RLD0,tαu(t)| t=tk+12=τ−α∑ℓ=0kϖℓ(α)u(tk−ℓτ)+O(τ2),k=0,1,…,α∈(0,1), where the coefficients $\varpi_{\ell}^{(\alpha)}$$(\ell=0,1,\ldots,k)$ can be determined via the following generating function G(z)=(3α+12α−2α+1αz+α+12αz2)α,|z|<1. Next, applying the formula to the time fractional Cable equations with Riemann–Liouville derivative in one and two space dimensions. Then the high-order compact finite difference schemes are obtained. The solvability, stability and convergence with orders $\mathcal{O}(\tau^2+h^4)$ and $\mathcal{O}(\tau^2+h_x^4+h_y^4)$ are shown, where $\tau$ is the temporal stepsize and $h$, $h_x$, $h_y$ are the spatial stepsizes, respectively. Finally, numerical experiments are provided to support the theoretical analysis.

Analytical Approach for the Space-time Nonlinear Partial Differential Fractional Equation

2014 ◽  
Vol 15 (7-8) ◽  
Author(s):  
Ahmet Bekir ◽  
Özkan Güner
Keyword(s):  
Function Method ◽  
Analytical Approach ◽  
Fractional Derivatives ◽  
Space Time ◽  
Variable Method ◽  
Partial Differential ◽  
Functional Variable ◽  
Liouville Derivative ◽  
Functional Variable Method ◽  
Fractional Equation

AbstractIn this paper, the fractional derivatives in the sense of modified Riemann–Liouville derivative and the functional variable method, exp-function method and

scholarly journals On the influence of the method of approximation of an unknown function on the stability of numerical methods for solving the anomalous diffusion equation

2019 ◽  
pp. 23-29
Author(s):  
Vladimir Andreevich Litvinov
Keyword(s):  
Diffusion Equation ◽  
Numerical Solution ◽  
Anomalous Diffusion ◽  
Caputo Derivative ◽  
Numerical Algorithms ◽  
Difference Approximation ◽  
Liouville Derivative ◽  
The Subject ◽  
The Difference ◽  
The Stability

The subject of the research is numerical algorithms for solving fractional partial differential equations. The object of the study is the stability of several algorithms for the numerical solution of the anomalous diffusion equation. Algorithms based on the difference representation of the fractional Riemann-Liuville derivative and the Caputo derivative for various orders of accuracy are considered. A comparison is made of the results of numerical calculations using the analyzed algorithms for a model problem with the exact solution of the anomalous diffusion equation for various orders of the fractional derivative with respect to the spatial coordinate. The results of the work were obtained on the basis of the analysis of the constructed difference schemes for stability, the conducted numerical experiments and a comparative analysis of the data obtained. The main conclusions of the study are the advantage of using the approximation of the fractional Caputo derivative compared to using the difference scheme for the fractional Riemann-Liouville derivative in the numerical solution of the anomalous diffusion equation. The paper also indicates the importance of choosing the method of difference approximation of the second derivative, which is a derivative of the Caputo.

scholarly journals Optimal Control Problem Investigation for Linear Time-Invariant Systems of Fractional Order with Lumped Parameters Described by Equations with Riemann-Liouville Derivative

2016 ◽  
Vol 2016 ◽  
pp. 1-12 ◽  
Author(s):  
V. A. Kubyshkin ◽  
S. S. Postnov
Keyword(s):  
Fractional Order ◽  
Caputo Derivative ◽  
Linear Time ◽  
Fractional Order Systems ◽  
Linear Time Invariant ◽  
Liouville Derivative ◽  
Lumped Parameters ◽  
Linear Time Invariant Systems ◽  
Time Invariant ◽  
Problem Setting

This paper studies two optimal control problems for linear time-invariant systems of fractional order with lumped parameters whose dynamics is described by equations which contain Riemann-Liouville derivative. The first problem is to find control with minimal norm and the second one is to find control with minimal control time at given restriction for control norm. The problem setting with nonlocal initial conditions is considered which differs from other known settings for integer-order systems and fractional-order systems described in terms of equations with Caputo derivative. Admissible controls are allowed to belong to the class of functions which arep-integrable on half segment. The basic investigation approach is the moment method. The correctness and solvability of moment problem are validated for considered problem setting for the system of arbitrary dimension. It is shown that corresponding conditions are analogous to those derived for systems which are described in terms of equations with Caputo derivative. For several particular cases of one- and two-dimensional systems the posed problems are solved explicitly. The dependencies of basic values from derivative index and control time are analyzed. The comparison is performed of obtained results with known results for analogous integer-order systems and fractional-order systems which are described by equations with Caputo derivative.

Application of a Homogeneous Balance Method to Exact Solutions of Nonlinear Fractional Evolution Equations

2014 ◽  
Vol 9 (2) ◽  
Author(s):  
H. Jafari ◽  
H. Tajadodi ◽  
D. Baleanu
Keyword(s):  
Exact Solutions ◽  
Fractional Derivatives ◽  
Evolution Equations ◽  
Telegraph Equation ◽  
Fisher Equation ◽  
Fractional Evolution Equations ◽  
Liouville Derivative ◽  
Nonlinear Telegraph Equation ◽  
Homogeneous Balance Method ◽  
Homogeneous Balance

The fractional Fan subequation method of the fractional Riccati equation is applied to construct the exact solutions of some nonlinear fractional evolution equations. In this paper, a powerful algorithm is developed for the exact solutions of the modified equal width equation, the Fisher equation, the nonlinear Telegraph equation, and the Cahn–Allen equation of fractional order. Fractional derivatives are described in the sense of the modified Riemann–Liouville derivative. Some relevant examples are investigated.

Sign in / Sign up

Export Citation Format

Share Document